Decorrelation of Edge Plasma Turbulence at the Transition from
Low- to High-Con nement Mode in the DIII{D Tokamak
Coda, S.
, Porkolab, M.
Massachusetts Institute of Technology Plasma Science and Fusion Center
Cambridge, MA 02139 USA Burrell, K.H. General Atomics P.O. Box 85608 San Diego, CA 92186-5608
ABSTRACT
The modi cation of turbulence in the edge plasma of the DIII{D tokamak at the tran-sition from the low to the high mode of con nement is investigated with a phase-contrast imaging diagnostic. The amplitude and radial correlation length of the turbulence in the con nement region decrease at the transition, whereas the decorrelation time increases. The transition model of Biglari, Diamond, and Terry [Phys. Fluids B 2, 1 (1990)], based
on turbulence decorrelation by EB velocity shear, is quantitatively substantiated by
measurements of the theoretical control parameter. Further quantitative predictions of the theory are tested for the rst time.
A central goal of research in nuclear fusion by magnetic con nement has been the minimization of anomalous plasma transport, de ned as all transport in excess of the in-escapable rate due to Coulomb collisions. Indeed, a great deal of progress has been made over the past few decades, which have seen the successive discoveries of various regimes of improved con nement [1]. Much current research aims at improving the understand-ing and control of these regimes. Anomalous transport is widely attributed to turbulent uctuations caused by microinstabilities [1]. However, the experimental evidence for this connection is still incomplete, and much remains unknown about the nature and proper-ties of the turbulence and of the underlying instabiliproper-ties. More generally, understanding turbulence is one of the remaining great scienti c challenges of classical physics.
The rst and most universal of the enhanced con nement regimes is the high-con nement mode (H{mode) [2]. There is now considerable evidence that a single mecha-nism, turbulence decorrelation and suppression by a sheared EBvelocity, is responsible
for the transition from the low-con nement mode (L{mode) to the H{mode [3]. This same mechanism is thought to play an important role in the enhanced-con nement regimes dis-covered more recently [4]. A theoretical model for the shear decorrelation mechanism was proposed nearly a decade ago [5] and by now has become widely accepted, with additional support from recent numerical simulations [6]. Measurements by electrostatic probes [7] and re ectometry [8] have provided initial corroboration of the theory.
An experimental test of the predictions of the model requires the simultaneous mea-surement with good spatial and temporal resolution of the correlation length and decor-relation time of the turbulence, as well as of the electric eld, in the plasma edge. In this Letter, we present the rst simultaneous, time-resolved, noninvasive measurement of these correlation parameters across the L{H transition. The measurement was performed with a high-spatial-resolution phase-contrast imaging (PCI) diagnostic [9] in the DIII{D toka-mak [10]. Used in conjunction with measurements of the electric eld by charge-exchange recombination (CER) spectroscopy [11], the PCI data permit a quantitative test of the the-oretical criterion for shear decorrelation, which is found to be well satis ed. In addition, we nd that the radial correlation length, as well as the amplitude, of the turbulent density uctuations in the con nement region decrease at the transition, whereas the decorrelation time increases. These measurements allow us to test for the rst time quantitative pre-dictions generated by two versions of the theory [5,12] for the changes in those correlation quantities. Incomplete agreement is obtained in both cases, suggesting that re nements of the theory are desirable in order to achieve a more complete understanding of the L{H transition.
The inception of the H{mode, related to the attainment of a heating power threshold, is generally abrupt and is characterized by the formation of a transport barrier in the plasma edge, concomitant with a reduction in the edge turbulence level. The L{H tran-sition is empirically identi ed by a decrease in the spectroscopic D edge emission, which
indicates reduced edge particle transport. On a slower time scale, the con nement im-provement extends to the core [1,4]. During the evolution of the H{mode, Edge Localized Modes (ELMs) { quasiperiodic edge instabilities { are often observed. In the remainder of this Letter we shall only consider the early, ELM-free H{mode phase occupying the rst few tens of milliseconds after the L{H transition. The properties of turbulence change substantially in the later phase, especially in the presence of ELMs.
The PCI diagnostic [13] employs a 20-W CO2 laser beam of 10.6
m wavelength
and 7.6 cm width, which is launched vertically through the outer plasma edge of DIII{D and is then passed through a Zernike phase-contrast lter [14] to create a radial image of the plasma onto a 16-element HgMnTe linear detector array. Thus, the system measures the vertical line integrals of the density uctuations at 16 radial locations. Any eight of the 16 signals can be collected in a given plasma discharge. The radial resolution is 0.5 cm, the temporal resolution is 0.5 s, and the density sensitivity is 10
15 m,3. The
frequency range is 8{1000 kHz, the lower limit being imposed by mechanical vibrations; the wave-number range is 0.65{6 cm,1. The vertical integration combined with the eld-line
geometry at the edge constrains the measured wave vectors to be essentially radial, i.e., perpendicular to the magnetic ux surfaces; as a result, the measured frequency spectra experience negligible Doppler shifts in response to the poloidal and toroidalEBdrifts due
to the radial electric eld, as con rmed by extensive analytical and numerical modeling [13,15].
Experimental tests of theory require the estimation of the correlation length and decorrelation time of turbulent plasma eddies [5]. Since an eddy travels at the group velocity of the uctuations, a Lagrangian rather than a Eulerian approach is required to determine its true lifetime. To this e ect we calculate the autocorrelation function of the peak spatial Fourier component of the turbulence. The intrinsic eddy decorrelation time is taken to be the e-folding time of this function [13] and is systematically larger than the
stationary (Eulerian) decorrelation time, as expected. This approach is contingent on the lack of Doppler broadening. The radial correlation length is estimated as the e-folding
length of the envelope of the equal-time spatial correlation coecient.
The properties of turbulence in L{ and H{mode have been studied with the PCI system in detail in over 100 DIII{D diverted deuterium discharges, in the region from 6 cm inside to 4 cm outside the last closed ux surface (LCFS). Several quantitative features exhibit signi cant dependences on plasma parameters, which are beyond the scope of this Letter [13]. Here we shall focus on the systematic changes occurring at the L{H transition. Inside the LCFS, the signal amplitude is lower in H{mode than in L{mode. The reduction factor is largest in the region just inside the LCFS and ranges from 1.2 to 3 in our data set; further into the plasma, 4{6 cm inside the LCFS, that factor is generally close to 1. In the region of open eld lines (scrape-o layer, or SOL), the H{mode amplitude can be slightly smaller than its L{mode counterpart, but it is also often larger by up to a factor of two. Representative radial pro les of the density uctuations, obtained by using a set of similar discharges with di erent positions of the LCFS, are shown in Fig. 1. The peak located 0.5{1.5 cm inside the LCFS is a universal feature. It must be remembered that these data represent the rms values of the line-integrated density uctuations; thus, part of the signal reduction in H{mode could be due to a decrease in the vertical correlation length.
The frequency spectrum is monotonically decreasing everywhere in L{mode, generally with an approximate !
,2 dependence. In H{mode, the spectrum is broader overall than
in L{mode but exhibits irregular multiple peaks. These features are re ected in the H{ mode autocorrelation function, which is narrower at small delay times but exhibits a more prominent tail at large delays than its L{mode counterpart. The corresponding decorrelation time is generally larger in H{mode than in L{mode in the region from the
LCFS to 1.5 cm inside, smaller elsewhere; its value in all cases varies between 10 and 50
s. The intrinsic eddy decorrelation time, while systematically larger, displays the same
relative phenomenology: typical values in the region just inside the LCFS are 20{40s in
L{mode, 40{80 s in H{mode. The inaccessible frequency components below 8 kHz could
further increase these values: hence, these times should be taken as lower limits.
The radial correlation length decreases at the L{H transition inside the LCFS, while it remains constant or increases, sometimes substantially, in the SOL. The reduction factor in the con nement region ranges from 1.3 to 3, with values from 1.2 to 3.5 cm in L{mode and from 0.4 to 1.2 cm in H{mode, depending on plasma parameters. In the SOL, the correlation length is generally shorter than 1 cm in L{mode, while values up to 3 cm have been documented in H{mode [13].
The uctuation suppression always appears to be simultaneous with the L{H transi-tion, as de ned by the start of the decrease in the D signal, and occurs in less than 0.1
ms. The integration time for the estimation of the correlation parameters is dictated by statistical requirements, and is typically 5 ms. Within this resolution, the change in the correlation length is also seen immediately after the transition, i.e., in the rst interval lying entirely in H{mode. The changes in the decorrelation time are more erratic. This is shown in Fig. 2 for the case of a dithering transition.
The reduction in transport is believed to be caused by sheared poloidal EB ows
inducing decorrelation between di erentially rotating eddies [5]. This causal relation has been corroborated by CER measurements of the edge radial electric eld with 0.5-cm resolution in DIII{D, which have shown an increase in the E shear just before the L{H
transition [16]; the eld forms a well structure with a minimum just inside the LCFS. The analysis of the EB shear decorrelation e ect, carried out initially by Biglari, Diamond,
and Terry (BDT) [5] in cylindrical geometry, was later extended by the full toroidal for-mulation of Hahm and Burrell (HB) [12]. According to this analysis, the H{mode state is characterized by the condition !
s ! T, where ! s v 0 EB L r =L
is the shearing rate, v 0 EB = ( R B =B ) @[E r =(R B )]
=@r, R is the major radius, r is the minor radius, B and B
are the poloidal and toroidal magnetic elds respectively, L
r and L
are the radial
and poloidal correlation lengths respectively, ! T
1= d, and
d is the eddy decorrelation
time; the turbulence quantities here refer to the state with no E shear (L{mode). This
asymptotic criterion has been widely accepted and has appeared as a key ingredient in a number of other theories [1]. With L
r and
d provided by PCI and v
0
EB supplied by
CER, the greatest uncertainty in a test of the inequality comes from a lack of accurate measurements of L
; from far-infrared scattering spectra [17], we estimate L
= 3
1 cm
[18].
Figure 3 shows a comparison of! s and
!
T in L{ and H{mode in the region where the
transport barrier is created. Clearly the condition ! s
!
T is well satis ed in H{mode,
whereas the two quantities are comparable in L{mode, as should be expected for the phase leading up to the transition. These statements hold true for all the discharges to which this test has been applied. As was mentioned before, it is possible that!
T is overestimated
by the lack of access to the lowest frequencies; if this were the case, the inequality would be satis ed to an even greater degree. We emphasize again that these results apply to the L{H transition; there are indications that in the fully evolved H{mode other e ects, such as ow curvature and dephasing between potential and density uctuations [16], are at
least partly responsible for the continued low rate of transport.
As a result of the di erent formalisms employed by the BDT and HB models (cylin-drical and ballooning, respectively), di erent predictions are obtained for the changes in the radial correlation length and in the decorrelation time from L{ to H{mode as func-tions of the H{mode shearing rate. The BDT formulas are L
rH =L rL ' (2! sH dL) ,1=3 and dH = dL ' (2! sH dL)
,2=3; the HB formulas are L rH =L rL ' (1 + ! 2 sH 2 dL) ,1=2 and dH = dL
'1 (neglecting residual logarithmic dependences). A test of these predictions on
a representative set of discharges is shown in Fig. 4. It is apparent that the BDT theory provides a better t to the spatial correlation data, whereas the HB theory is a better pre-dictor of the temporal correlation data. These incomplete agreements may be attributed to the non-self-consistent nature of both models, which do not consider the reaction of the turbulence on the ows and, through the modi cation of transport, on the plasma parameters. While the evidence strongly supports the identi cation of !
s =!
T as the
con-trol parameter for the L{H transition, the detailed changes occurring in the turbulence spectrum can be expected to be a ected by the rapid variations in plasma conditions, par-ticularly the pressure gradient, that occur at the transition. These variations are ignored by the theories considered here. Conversely, in self-consistent theories [19] the decorrela-tion time has been modeled only in an ad hoc manner, with a dependence
d /!
,2
s which
is even less satisfactory than the BDT prediction. Indeed,
d is experimentally found to
increase at the L{H transition. This may simply indicate an increased stability of the modes, which leads to a smaller growth rate and thus to a larger decorrelation time; or it could be argued that the chief L{mode instabilities (e.g., ion-temperature-gradient modes [1]) are drastically quenched in H{mode and are replaced by new and less virulent insta-bilities [16]. In this case, a proper test of the predictions discussed above would require an isolated analysis of the modes that are suppressed, a very dicult task in practice.
A complete self-consistent theory of the L{H transition would also require an explicit calculation of the anomalous di usivity. Only heuristic estimates can be used at present. In a random-walk model, the di usivity D
rw = L
2
r =
d calculated from the data of Fig. 4
decreases by a factor ranging from 2.5 to 10 from L{ to H{mode, more than the twofold increase typically observed in the con nement time. However, it must be remembered that the PCI measurements are performed at the edge, where the transport barrier forms rst, while the con nement time is a global quantity. In addition, turbulence suppression is known to be stronger at the outer than at the inner edge [8]. It can thus be argued that the local di usivity (a quantity that is dicult to measure accurately at the edge) may decrease by more than a factor of two, as suggested by the random-walk estimate.
In conclusion, the rst nonperturbing, simultaneous measurement of the radial cor-relation length and decorcor-relation time of the turbulence with high temporal and spatial resolution has been carried out in the edge plasma of the DIII{D tokamak. The results pre-sented here are quantitatively consistent with the theoretical prediction that theEB ow
shearing rate should greatly exceed the intrinsic turbulence decorrelation rate in H{mode and that the two rates should be comparable just before the L{ to H{mode transition. This supports the paradigm of turbulence decorrelation by a shearedEB ow producing
the improved-con nement, H{mode state [5,12]. However, comparison ofL r and
d before
and after the L{H transition shows that only some of the detailed quantitative predictions of the simple analytic theories [5,12] are correct. This indicates the need for more
plete theories which self-consistently include the e ect of the drastic changes in background plasma conditions that occur after the transition.
The authors are grateful to the DIII{D team for the operation of the tokamak, heating systems, and diagnostics, and to B. A. Carreras, P. H. Diamond, and T. S. Hahm for illuminating discussions. This work was supported by the U.S. Department of Energy under Grant No. DE-FG02-91ER54109 and Contract No. DE-AC03-89ER51114.
REFERENCES
* Present and permanent address: Centre de Recherches en Physique des Plasmas, Ecole Polytechnique Federale de Lausanne, CH{1015 Lausanne, Switzerland.
Email address: Stefano.Coda@ep .ch
[1] See B. A. Carreras, IEEE Trans. Plasma Sci. 25, 1281 (1997), and references therein.
[2] F. Wagner et al., Phys. Rev. Lett. 49, 1408 (1982).
[3] K. H. Burrell et al., Phys. Plasmas1, 1536 (1994).
[4] K. H. Burrell, Phys. Plasmas 4, 1499 (1997).
[5] H. Biglari, P. H. Diamond, and P. W. Terry, Phys. Fluids B 2, 1 (1990).
[6] Z. Lin et al., Science 281, 1835 (1998).
[7] Ch. P. Ritzet al., Phys. Rev. Lett. 65, 2543 (1990); G.R. Tynanet al., Phys. Plasmas 1, 3301 (1994); S. Ohdachi et al., Plasma Phys. Control. Fusion 36, A201 (1994).
[8] E. J. Doyle et al., in Proceedings of the 14th International Conference on Plasma Physics and Controlled Nuclear Fusion Research, Wurzburg, 1992 (International Atomic Energy Agency, Vienna, Austria, 1993), vol. 1, p. 235.
[9] S. Coda, M. Porkolab, and T. N. Carlstrom, Rev. Sci. Instrum. 63, 4974 (1992).
[10] J. L. Luxon and L. G. Davis, Fusion Technol. 8, 441 (1985).
[11] P. Gohil et al., in Proceedings of the 14th IEEE/NPSS Symposium on Fusion Engi-neering, San Diego, 1991 (IEEE, New York, NY, 1992), Vol. 2, p. 1199.
[12] T. S. Hahm and K. H. Burrell, Phys. Plasmas 2, 1648 (1995).
[13] S. Coda, Ph.D. Thesis, Massachusetts Institute of Technology, 1997. [14] F. Zernike, Physica 1, 689 (1934).
[15] S. Coda and M. Porkolab, Rev. Sci. Instrum. 66, 454 (1995).
[16] R. A. Moyer et al., Phys. Plasmas2, 2397 (1995).
[17] C. L. Rettig, Ph.D. Thesis, University of California, Los Angeles, 1993.
[18] This estimate is consistent with a majority of measurements in tokamak edges placing the value of L
=L
r between 1 and 3. See, e.g., Ch. P. Ritz et al., Phys. Fluids 27,
2956 (1984); S. J. Zweben and R. W. Gould, Nucl. Fusion 25, 171 (1985); R. J. Fonck
et al., Plasma Phys. Control. Fusion34, 1993 (1992).
[19] P. H. Diamondet al., Phys. Rev. Lett. 72, 2565 (1994).
FIGURES
FIG. 1. Spatial distribution of rms values of line integrals of density uctuations in L{ and H{mode, as a function of horizontal distance from LCFS on the midplane. Data are collected from a set of similar shots with varying LCFS position (magnetic eld B
=2.1
T, current I
p=1.35 MA, line-averaged density n
e=4 10
19 m,3, input neutral-beam power
1.1{2.3 MW). The error bars shown re ect chord-to-chord calibration uncertainties; the common absolute uncertainty is 40%.
FIG. 2. Evolution across the L{H transition of (a) D emission, (b) rms value of
line-integrated density uctuations (40% uncertainty on the absolute value), (c) radial correlation length, and (d) intrinsic eddy decorrelation time. Measurements (b){(d) are made with PCI in the region from the LCFS to 0.8 cm inside it. Plasma parameters:
B =2.1 T, I p=1.25 MA, n e=3.5 10
19 m,3, input neutral-beam power 6.2 MW.
FIG. 3. Comparison of shearing rate and turbulence decorrelation rate in L{ and H{ mode (25 ms before and after the transition, respectively). The decorrelation rate is shown as a segment over the spatial range sampled by PCI to obtain its value. The abscissa is a normalized radial coordinate (=1 at the LCFS). Plasma parameters: B
=2.16 T, I p=1.54 MA, n e=3 10
19 m,3, input neutral-beam power 8.5 MW.
FIG. 4. Ratios of (a) radial correlation lengths and (b) decorrelation times in H{ and L{mode, as functions of the H{mode shearing rate: experimental values averaged in the region of normalized poloidal ux 0.95{1 (open symbols), BDT predictions (solid lines), HB predictions (dashed lines).
L
L
L
